儚い describes something fragile and destined to disappear, such as a dream, love, or a brief moment. It has a poetic nuance that is stronger than simply saying "short-lived."
In the song, the singer realizes that people are more fragile and fleeting than he had imagined: 自分で思ってたよりも もろく儚い人間で. This memorable word captures the sadness and impermanence at the heart of the breakup.
"Still" opens with a list of routines. The singer still has the same fights, the same quick kisses, the same sleepy mornings: "いつもの様にすぐにケンカをして" (as usual, we picked a fight right away). Then the fourth line breaks the pattern: "いつもと違う君がいない僕のそばに" (what's different is that you're not beside me). The phrase いつもの様に ("like always") repeats three times, so you hear the habit carry on before you hear what's missing.
The most painful scene is small. Before dawn the two of them lie in bed hearing only the clock, and when she walks out, he asks himself "なぜ寝てるフリしたんだろ" (why did I pretend to be asleep?). He nearly dials her number, "かけ慣れた No" (the number I'm used to calling), and then "静かに指を止めた" (quietly stopped my finger). Later he starts looking for her way home and gives up with a sigh. Every attempt to fix things stops halfway.
The chorus switches to English: "Still I love you never fade." Then comes a run of lines that all start with 忘れたいから (because I want to forget): acting normal, throwing out her photos, drinking until he passes out. The last one turns it around with のに (even though): "忘れたいのに気づけば君の事ばかり" (I want to forget, yet before I know it, it's all about you). That one-word change from から to のに is a neat grammar lesson.
In the end he leaves the city because "どこにでも君が溢れて" (you overflow everywhere). He never manages to say "今ならまだ" (it's not too late yet), and the song closes on "さよならさよなら愛した人" (goodbye, goodbye, the person I loved). Two goodbyes and nothing more. It's blunt, and it works.